3.385 \(\int \frac {1}{a+b \tan ^4(c+d x)} \, dx\)

Optimal. Leaf size=302 \[ \frac {\sqrt [4]{b} \left (\sqrt {a}-\sqrt {b}\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} d (a+b)}-\frac {\sqrt [4]{b} \left (\sqrt {a}-\sqrt {b}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \tan (c+d x)}{\sqrt [4]{a}}+1\right )}{2 \sqrt {2} a^{3/4} d (a+b)}-\frac {\sqrt [4]{b} \left (\sqrt {a}+\sqrt {b}\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \tan (c+d x)+\sqrt {a}+\sqrt {b} \tan ^2(c+d x)\right )}{4 \sqrt {2} a^{3/4} d (a+b)}+\frac {\sqrt [4]{b} \left (\sqrt {a}+\sqrt {b}\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \tan (c+d x)+\sqrt {a}+\sqrt {b} \tan ^2(c+d x)\right )}{4 \sqrt {2} a^{3/4} d (a+b)}+\frac {x}{a+b} \]

[Out]

x/(a+b)+1/4*b^(1/4)*arctan(1-b^(1/4)*2^(1/2)*tan(d*x+c)/a^(1/4))*(a^(1/2)-b^(1/2))/a^(3/4)/(a+b)/d*2^(1/2)-1/4
*b^(1/4)*arctan(1+b^(1/4)*2^(1/2)*tan(d*x+c)/a^(1/4))*(a^(1/2)-b^(1/2))/a^(3/4)/(a+b)/d*2^(1/2)-1/8*b^(1/4)*ln
(a^(1/2)-a^(1/4)*b^(1/4)*2^(1/2)*tan(d*x+c)+b^(1/2)*tan(d*x+c)^2)*(a^(1/2)+b^(1/2))/a^(3/4)/(a+b)/d*2^(1/2)+1/
8*b^(1/4)*ln(a^(1/2)+a^(1/4)*b^(1/4)*2^(1/2)*tan(d*x+c)+b^(1/2)*tan(d*x+c)^2)*(a^(1/2)+b^(1/2))/a^(3/4)/(a+b)/
d*2^(1/2)

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Rubi [A]  time = 0.32, antiderivative size = 302, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 9, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.643, Rules used = {3661, 1171, 203, 1168, 1162, 617, 204, 1165, 628} \[ \frac {\sqrt [4]{b} \left (\sqrt {a}-\sqrt {b}\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} d (a+b)}-\frac {\sqrt [4]{b} \left (\sqrt {a}-\sqrt {b}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \tan (c+d x)}{\sqrt [4]{a}}+1\right )}{2 \sqrt {2} a^{3/4} d (a+b)}-\frac {\sqrt [4]{b} \left (\sqrt {a}+\sqrt {b}\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \tan (c+d x)+\sqrt {a}+\sqrt {b} \tan ^2(c+d x)\right )}{4 \sqrt {2} a^{3/4} d (a+b)}+\frac {\sqrt [4]{b} \left (\sqrt {a}+\sqrt {b}\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \tan (c+d x)+\sqrt {a}+\sqrt {b} \tan ^2(c+d x)\right )}{4 \sqrt {2} a^{3/4} d (a+b)}+\frac {x}{a+b} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Tan[c + d*x]^4)^(-1),x]

[Out]

x/(a + b) + ((Sqrt[a] - Sqrt[b])*b^(1/4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Tan[c + d*x])/a^(1/4)])/(2*Sqrt[2]*a^(3/4
)*(a + b)*d) - ((Sqrt[a] - Sqrt[b])*b^(1/4)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Tan[c + d*x])/a^(1/4)])/(2*Sqrt[2]*a^(
3/4)*(a + b)*d) - ((Sqrt[a] + Sqrt[b])*b^(1/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Tan[c + d*x] + Sqrt[b]*Ta
n[c + d*x]^2])/(4*Sqrt[2]*a^(3/4)*(a + b)*d) + ((Sqrt[a] + Sqrt[b])*b^(1/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1
/4)*Tan[c + d*x] + Sqrt[b]*Tan[c + d*x]^2])/(4*Sqrt[2]*a^(3/4)*(a + b)*d)

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 1162

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(2*d)/e, 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 1165

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(-2*d)/e, 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rule 1168

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[a*c, 2]}, Dist[(d*q + a*e)/(2*a*c),
Int[(q + c*x^2)/(a + c*x^4), x], x] + Dist[(d*q - a*e)/(2*a*c), Int[(q - c*x^2)/(a + c*x^4), x], x]] /; FreeQ
[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && NegQ[-(a*c)]

Rule 1171

Int[((d_) + (e_.)*(x_)^2)^(q_)/((a_) + (c_.)*(x_)^4), x_Symbol] :> Int[ExpandIntegrand[(d + e*x^2)^q/(a + c*x^
4), x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && IntegerQ[q]

Rule 3661

Int[((a_) + (b_.)*((c_.)*tan[(e_.) + (f_.)*(x_)])^(n_))^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x]
, x]}, Dist[(c*ff)/f, Subst[Int[(a + b*(ff*x)^n)^p/(c^2 + ff^2*x^2), x], x, (c*Tan[e + f*x])/ff], x]] /; FreeQ
[{a, b, c, e, f, n, p}, x] && (IntegersQ[n, p] || IGtQ[p, 0] || EqQ[n^2, 4] || EqQ[n^2, 16])

Rubi steps

\begin {align*} \int \frac {1}{a+b \tan ^4(c+d x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{\left (1+x^2\right ) \left (a+b x^4\right )} \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {1}{(a+b) \left (1+x^2\right )}+\frac {b-b x^2}{(a+b) \left (a+b x^4\right )}\right ) \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\tan (c+d x)\right )}{(a+b) d}+\frac {\operatorname {Subst}\left (\int \frac {b-b x^2}{a+b x^4} \, dx,x,\tan (c+d x)\right )}{(a+b) d}\\ &=\frac {x}{a+b}-\frac {\left (1-\frac {\sqrt {b}}{\sqrt {a}}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {a} \sqrt {b}+b x^2}{a+b x^4} \, dx,x,\tan (c+d x)\right )}{2 (a+b) d}+\frac {\left (1+\frac {\sqrt {b}}{\sqrt {a}}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {a} \sqrt {b}-b x^2}{a+b x^4} \, dx,x,\tan (c+d x)\right )}{2 (a+b) d}\\ &=\frac {x}{a+b}-\frac {\left (1-\frac {\sqrt {b}}{\sqrt {a}}\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx,x,\tan (c+d x)\right )}{4 (a+b) d}-\frac {\left (1-\frac {\sqrt {b}}{\sqrt {a}}\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx,x,\tan (c+d x)\right )}{4 (a+b) d}-\frac {\left (\left (\sqrt {a}+\sqrt {b}\right ) \sqrt [4]{b}\right ) \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{b}}+2 x}{-\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx,x,\tan (c+d x)\right )}{4 \sqrt {2} a^{3/4} (a+b) d}-\frac {\left (\left (\sqrt {a}+\sqrt {b}\right ) \sqrt [4]{b}\right ) \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{b}}-2 x}{-\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx,x,\tan (c+d x)\right )}{4 \sqrt {2} a^{3/4} (a+b) d}\\ &=\frac {x}{a+b}-\frac {\left (\sqrt {a}+\sqrt {b}\right ) \sqrt [4]{b} \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \tan (c+d x)+\sqrt {b} \tan ^2(c+d x)\right )}{4 \sqrt {2} a^{3/4} (a+b) d}+\frac {\left (\sqrt {a}+\sqrt {b}\right ) \sqrt [4]{b} \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \tan (c+d x)+\sqrt {b} \tan ^2(c+d x)\right )}{4 \sqrt {2} a^{3/4} (a+b) d}-\frac {\left (\left (\sqrt {a}-\sqrt {b}\right ) \sqrt [4]{b}\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{b} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} (a+b) d}+\frac {\left (\left (\sqrt {a}-\sqrt {b}\right ) \sqrt [4]{b}\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{b} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} (a+b) d}\\ &=\frac {x}{a+b}+\frac {\left (\sqrt {a}-\sqrt {b}\right ) \sqrt [4]{b} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} (a+b) d}-\frac {\left (\sqrt {a}-\sqrt {b}\right ) \sqrt [4]{b} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} (a+b) d}-\frac {\left (\sqrt {a}+\sqrt {b}\right ) \sqrt [4]{b} \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \tan (c+d x)+\sqrt {b} \tan ^2(c+d x)\right )}{4 \sqrt {2} a^{3/4} (a+b) d}+\frac {\left (\sqrt {a}+\sqrt {b}\right ) \sqrt [4]{b} \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \tan (c+d x)+\sqrt {b} \tan ^2(c+d x)\right )}{4 \sqrt {2} a^{3/4} (a+b) d}\\ \end {align*}

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Mathematica [A]  time = 0.57, size = 228, normalized size = 0.75 \[ \frac {8 a^{3/4} \tan ^{-1}(\tan (c+d x))+\sqrt {2} \sqrt [4]{b} \left (2 \left (\sqrt {a}-\sqrt {b}\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \tan (c+d x)}{\sqrt [4]{a}}\right )-2 \left (\sqrt {a}-\sqrt {b}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \tan (c+d x)}{\sqrt [4]{a}}+1\right )-\left (\sqrt {a}+\sqrt {b}\right ) \left (\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \tan (c+d x)+\sqrt {a}+\sqrt {b} \tan ^2(c+d x)\right )-\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \tan (c+d x)+\sqrt {a}+\sqrt {b} \tan ^2(c+d x)\right )\right )\right )}{8 a^{3/4} d (a+b)} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Tan[c + d*x]^4)^(-1),x]

[Out]

(8*a^(3/4)*ArcTan[Tan[c + d*x]] + Sqrt[2]*b^(1/4)*(2*(Sqrt[a] - Sqrt[b])*ArcTan[1 - (Sqrt[2]*b^(1/4)*Tan[c + d
*x])/a^(1/4)] - 2*(Sqrt[a] - Sqrt[b])*ArcTan[1 + (Sqrt[2]*b^(1/4)*Tan[c + d*x])/a^(1/4)] - (Sqrt[a] + Sqrt[b])
*(Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Tan[c + d*x] + Sqrt[b]*Tan[c + d*x]^2] - Log[Sqrt[a] + Sqrt[2]*a^(1/4)
*b^(1/4)*Tan[c + d*x] + Sqrt[b]*Tan[c + d*x]^2])))/(8*a^(3/4)*(a + b)*d)

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fricas [B]  time = 0.74, size = 1541, normalized size = 5.10 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+tan(d*x+c)^4*b),x, algorithm="fricas")

[Out]

1/8*((a + b)*sqrt(((a^3 + 2*a^2*b + a*b^2)*d^2*sqrt(-(a^2*b - 2*a*b^2 + b^3)/((a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a
^4*b^3 + a^3*b^4)*d^4)) + 2*b)/((a^3 + 2*a^2*b + a*b^2)*d^2))*log((2*(a^3 - a*b^2)*d*sqrt(((a^3 + 2*a^2*b + a*
b^2)*d^2*sqrt(-(a^2*b - 2*a*b^2 + b^3)/((a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b^4)*d^4)) + 2*b)/((a^3 +
2*a^2*b + a*b^2)*d^2))*tan(d*x + c) + (a*b - b^2)*tan(d*x + c)^2 + a^2 - a*b + ((a^4 + 2*a^3*b + a^2*b^2)*d^2
*tan(d*x + c)^2 - (a^4 + 2*a^3*b + a^2*b^2)*d^2)*sqrt(-(a^2*b - 2*a*b^2 + b^3)/((a^7 + 4*a^6*b + 6*a^5*b^2 + 4
*a^4*b^3 + a^3*b^4)*d^4)))/(tan(d*x + c)^2 + 1)) - (a + b)*sqrt(((a^3 + 2*a^2*b + a*b^2)*d^2*sqrt(-(a^2*b - 2*
a*b^2 + b^3)/((a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b^4)*d^4)) + 2*b)/((a^3 + 2*a^2*b + a*b^2)*d^2))*lo
g(-(2*(a^3 - a*b^2)*d*sqrt(((a^3 + 2*a^2*b + a*b^2)*d^2*sqrt(-(a^2*b - 2*a*b^2 + b^3)/((a^7 + 4*a^6*b + 6*a^5*
b^2 + 4*a^4*b^3 + a^3*b^4)*d^4)) + 2*b)/((a^3 + 2*a^2*b + a*b^2)*d^2))*tan(d*x + c) - (a*b - b^2)*tan(d*x + c)
^2 - a^2 + a*b - ((a^4 + 2*a^3*b + a^2*b^2)*d^2*tan(d*x + c)^2 - (a^4 + 2*a^3*b + a^2*b^2)*d^2)*sqrt(-(a^2*b -
2*a*b^2 + b^3)/((a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b^4)*d^4)))/(tan(d*x + c)^2 + 1)) + (a + b)*sqrt
(-((a^3 + 2*a^2*b + a*b^2)*d^2*sqrt(-(a^2*b - 2*a*b^2 + b^3)/((a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b^4
)*d^4)) - 2*b)/((a^3 + 2*a^2*b + a*b^2)*d^2))*log(-(2*(a^3 - a*b^2)*d*sqrt(-((a^3 + 2*a^2*b + a*b^2)*d^2*sqrt(
-(a^2*b - 2*a*b^2 + b^3)/((a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b^4)*d^4)) - 2*b)/((a^3 + 2*a^2*b + a*b
^2)*d^2))*tan(d*x + c) + (a*b - b^2)*tan(d*x + c)^2 + a^2 - a*b - ((a^4 + 2*a^3*b + a^2*b^2)*d^2*tan(d*x + c)^
2 - (a^4 + 2*a^3*b + a^2*b^2)*d^2)*sqrt(-(a^2*b - 2*a*b^2 + b^3)/((a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3
*b^4)*d^4)))/(tan(d*x + c)^2 + 1)) - (a + b)*sqrt(-((a^3 + 2*a^2*b + a*b^2)*d^2*sqrt(-(a^2*b - 2*a*b^2 + b^3)/
((a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b^4)*d^4)) - 2*b)/((a^3 + 2*a^2*b + a*b^2)*d^2))*log((2*(a^3 - a
*b^2)*d*sqrt(-((a^3 + 2*a^2*b + a*b^2)*d^2*sqrt(-(a^2*b - 2*a*b^2 + b^3)/((a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b
^3 + a^3*b^4)*d^4)) - 2*b)/((a^3 + 2*a^2*b + a*b^2)*d^2))*tan(d*x + c) - (a*b - b^2)*tan(d*x + c)^2 - a^2 + a*
b + ((a^4 + 2*a^3*b + a^2*b^2)*d^2*tan(d*x + c)^2 - (a^4 + 2*a^3*b + a^2*b^2)*d^2)*sqrt(-(a^2*b - 2*a*b^2 + b^
3)/((a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b^4)*d^4)))/(tan(d*x + c)^2 + 1)) + 8*x)/(a + b)

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giac [A]  time = 3.19, size = 354, normalized size = 1.17 \[ \frac {\frac {2 \, {\left (\left (a b^{3}\right )^{\frac {1}{4}} b^{2} - \left (a b^{3}\right )^{\frac {3}{4}}\right )} {\left (\pi \left \lfloor \frac {d x + c}{\pi } + \frac {1}{2} \right \rfloor + \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} + 2 \, \tan \left (d x + c\right )\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )\right )}}{\sqrt {2} a^{2} b^{2} + \sqrt {2} a b^{3}} + \frac {2 \, {\left (\left (a b^{3}\right )^{\frac {1}{4}} b^{2} - \left (a b^{3}\right )^{\frac {3}{4}}\right )} {\left (\pi \left \lfloor \frac {d x + c}{\pi } + \frac {1}{2} \right \rfloor + \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} - 2 \, \tan \left (d x + c\right )\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )\right )}}{\sqrt {2} a^{2} b^{2} + \sqrt {2} a b^{3}} + \frac {{\left (\left (a b^{3}\right )^{\frac {1}{4}} b^{2} + \left (a b^{3}\right )^{\frac {3}{4}}\right )} \log \left (\tan \left (d x + c\right )^{2} + \sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} \tan \left (d x + c\right ) + \sqrt {\frac {a}{b}}\right )}{\sqrt {2} a^{2} b^{2} + \sqrt {2} a b^{3}} - \frac {{\left (\left (a b^{3}\right )^{\frac {1}{4}} b^{2} + \left (a b^{3}\right )^{\frac {3}{4}}\right )} \log \left (\tan \left (d x + c\right )^{2} - \sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} \tan \left (d x + c\right ) + \sqrt {\frac {a}{b}}\right )}{\sqrt {2} a^{2} b^{2} + \sqrt {2} a b^{3}} + \frac {4 \, {\left (d x + c\right )}}{a + b}}{4 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+tan(d*x+c)^4*b),x, algorithm="giac")

[Out]

1/4*(2*((a*b^3)^(1/4)*b^2 - (a*b^3)^(3/4))*(pi*floor((d*x + c)/pi + 1/2) + arctan(1/2*sqrt(2)*(sqrt(2)*(a/b)^(
1/4) + 2*tan(d*x + c))/(a/b)^(1/4)))/(sqrt(2)*a^2*b^2 + sqrt(2)*a*b^3) + 2*((a*b^3)^(1/4)*b^2 - (a*b^3)^(3/4))
*(pi*floor((d*x + c)/pi + 1/2) + arctan(-1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) - 2*tan(d*x + c))/(a/b)^(1/4)))/(sqr
t(2)*a^2*b^2 + sqrt(2)*a*b^3) + ((a*b^3)^(1/4)*b^2 + (a*b^3)^(3/4))*log(tan(d*x + c)^2 + sqrt(2)*(a/b)^(1/4)*t
an(d*x + c) + sqrt(a/b))/(sqrt(2)*a^2*b^2 + sqrt(2)*a*b^3) - ((a*b^3)^(1/4)*b^2 + (a*b^3)^(3/4))*log(tan(d*x +
c)^2 - sqrt(2)*(a/b)^(1/4)*tan(d*x + c) + sqrt(a/b))/(sqrt(2)*a^2*b^2 + sqrt(2)*a*b^3) + 4*(d*x + c)/(a + b))
/d

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maple [A]  time = 0.21, size = 374, normalized size = 1.24 \[ \frac {b \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \ln \left (\frac {\tan ^{2}\left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {1}{4}} \tan \left (d x +c \right ) \sqrt {2}+\sqrt {\frac {a}{b}}}{\tan ^{2}\left (d x +c \right )-\left (\frac {a}{b}\right )^{\frac {1}{4}} \tan \left (d x +c \right ) \sqrt {2}+\sqrt {\frac {a}{b}}}\right )}{8 d \left (a +b \right ) a}+\frac {b \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \tan \left (d x +c \right )}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )}{4 d \left (a +b \right ) a}-\frac {b \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (-\frac {\sqrt {2}\, \tan \left (d x +c \right )}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )}{4 d \left (a +b \right ) a}-\frac {\sqrt {2}\, \ln \left (\frac {\tan ^{2}\left (d x +c \right )-\left (\frac {a}{b}\right )^{\frac {1}{4}} \tan \left (d x +c \right ) \sqrt {2}+\sqrt {\frac {a}{b}}}{\tan ^{2}\left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {1}{4}} \tan \left (d x +c \right ) \sqrt {2}+\sqrt {\frac {a}{b}}}\right )}{8 d \left (a +b \right ) \left (\frac {a}{b}\right )^{\frac {1}{4}}}-\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \tan \left (d x +c \right )}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )}{4 d \left (a +b \right ) \left (\frac {a}{b}\right )^{\frac {1}{4}}}+\frac {\sqrt {2}\, \arctan \left (-\frac {\sqrt {2}\, \tan \left (d x +c \right )}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )}{4 d \left (a +b \right ) \left (\frac {a}{b}\right )^{\frac {1}{4}}}+\frac {\arctan \left (\tan \left (d x +c \right )\right )}{d \left (a +b \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b*tan(d*x+c)^4),x)

[Out]

1/8/d*b/(a+b)*(1/b*a)^(1/4)/a*2^(1/2)*ln((tan(d*x+c)^2+(1/b*a)^(1/4)*tan(d*x+c)*2^(1/2)+(1/b*a)^(1/2))/(tan(d*
x+c)^2-(1/b*a)^(1/4)*tan(d*x+c)*2^(1/2)+(1/b*a)^(1/2)))+1/4/d*b/(a+b)*(1/b*a)^(1/4)/a*2^(1/2)*arctan(2^(1/2)/(
1/b*a)^(1/4)*tan(d*x+c)+1)-1/4/d*b/(a+b)*(1/b*a)^(1/4)/a*2^(1/2)*arctan(-2^(1/2)/(1/b*a)^(1/4)*tan(d*x+c)+1)-1
/8/d/(a+b)/(1/b*a)^(1/4)*2^(1/2)*ln((tan(d*x+c)^2-(1/b*a)^(1/4)*tan(d*x+c)*2^(1/2)+(1/b*a)^(1/2))/(tan(d*x+c)^
2+(1/b*a)^(1/4)*tan(d*x+c)*2^(1/2)+(1/b*a)^(1/2)))-1/4/d/(a+b)/(1/b*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/b*a)^(1
/4)*tan(d*x+c)+1)+1/4/d/(a+b)/(1/b*a)^(1/4)*2^(1/2)*arctan(-2^(1/2)/(1/b*a)^(1/4)*tan(d*x+c)+1)+1/d/(a+b)*arct
an(tan(d*x+c))

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maxima [A]  time = 0.86, size = 261, normalized size = 0.86 \[ -\frac {\frac {b {\left (\frac {2 \, \sqrt {2} {\left (\sqrt {a} - \sqrt {b}\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {b} \tan \left (d x + c\right ) + \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} + \frac {2 \, \sqrt {2} {\left (\sqrt {a} - \sqrt {b}\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {b} \tan \left (d x + c\right ) - \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} - \frac {\sqrt {2} {\left (\sqrt {a} + \sqrt {b}\right )} \log \left (\sqrt {b} \tan \left (d x + c\right )^{2} + \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \tan \left (d x + c\right ) + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {3}{4}}} + \frac {\sqrt {2} {\left (\sqrt {a} + \sqrt {b}\right )} \log \left (\sqrt {b} \tan \left (d x + c\right )^{2} - \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \tan \left (d x + c\right ) + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {3}{4}}}\right )}}{a + b} - \frac {8 \, {\left (d x + c\right )}}{a + b}}{8 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+tan(d*x+c)^4*b),x, algorithm="maxima")

[Out]

-1/8*(b*(2*sqrt(2)*(sqrt(a) - sqrt(b))*arctan(1/2*sqrt(2)*(2*sqrt(b)*tan(d*x + c) + sqrt(2)*a^(1/4)*b^(1/4))/s
qrt(sqrt(a)*sqrt(b)))/(sqrt(a)*sqrt(sqrt(a)*sqrt(b))*sqrt(b)) + 2*sqrt(2)*(sqrt(a) - sqrt(b))*arctan(1/2*sqrt(
2)*(2*sqrt(b)*tan(d*x + c) - sqrt(2)*a^(1/4)*b^(1/4))/sqrt(sqrt(a)*sqrt(b)))/(sqrt(a)*sqrt(sqrt(a)*sqrt(b))*sq
rt(b)) - sqrt(2)*(sqrt(a) + sqrt(b))*log(sqrt(b)*tan(d*x + c)^2 + sqrt(2)*a^(1/4)*b^(1/4)*tan(d*x + c) + sqrt(
a))/(a^(3/4)*b^(3/4)) + sqrt(2)*(sqrt(a) + sqrt(b))*log(sqrt(b)*tan(d*x + c)^2 - sqrt(2)*a^(1/4)*b^(1/4)*tan(d
*x + c) + sqrt(a))/(a^(3/4)*b^(3/4)))/(a + b) - 8*(d*x + c)/(a + b))/d

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mupad [B]  time = 15.04, size = 4038, normalized size = 13.37 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a + b*tan(c + d*x)^4),x)

[Out]

(2*atan(((((20*a*b^5 + 4*b^6 - ((((128*a^2*b^6 - 64*a*b^7 + 448*a^3*b^5 + 256*a^4*b^4 - (tan(c + d*x)*(512*a^2
*b^7 + 512*a^3*b^6 - 512*a^4*b^5 - 512*a^5*b^4)*1i)/(2*a + 2*b))*1i)/(2*a + 2*b) + tan(c + d*x)*(32*a*b^6 + 16
*b^7 - 240*a^2*b^5))*1i)/(2*a + 2*b))*1i)/(2*a + 2*b) - 6*b^5*tan(c + d*x))/(2*a + 2*b) - (((20*a*b^5 + 4*b^6
- ((((128*a^2*b^6 - 64*a*b^7 + 448*a^3*b^5 + 256*a^4*b^4 + (tan(c + d*x)*(512*a^2*b^7 + 512*a^3*b^6 - 512*a^4*
b^5 - 512*a^5*b^4)*1i)/(2*a + 2*b))*1i)/(2*a + 2*b) - tan(c + d*x)*(32*a*b^6 + 16*b^7 - 240*a^2*b^5))*1i)/(2*a
+ 2*b))*1i)/(2*a + 2*b) + 6*b^5*tan(c + d*x))/(2*a + 2*b))/(((((20*a*b^5 + 4*b^6 - ((((128*a^2*b^6 - 64*a*b^7
+ 448*a^3*b^5 + 256*a^4*b^4 - (tan(c + d*x)*(512*a^2*b^7 + 512*a^3*b^6 - 512*a^4*b^5 - 512*a^5*b^4)*1i)/(2*a
+ 2*b))*1i)/(2*a + 2*b) + tan(c + d*x)*(32*a*b^6 + 16*b^7 - 240*a^2*b^5))*1i)/(2*a + 2*b))*1i)/(2*a + 2*b) - 6
*b^5*tan(c + d*x))*1i)/(2*a + 2*b) + ((((20*a*b^5 + 4*b^6 - ((((128*a^2*b^6 - 64*a*b^7 + 448*a^3*b^5 + 256*a^4
*b^4 + (tan(c + d*x)*(512*a^2*b^7 + 512*a^3*b^6 - 512*a^4*b^5 - 512*a^5*b^4)*1i)/(2*a + 2*b))*1i)/(2*a + 2*b)
- tan(c + d*x)*(32*a*b^6 + 16*b^7 - 240*a^2*b^5))*1i)/(2*a + 2*b))*1i)/(2*a + 2*b) + 6*b^5*tan(c + d*x))*1i)/(
2*a + 2*b))))/(d*(2*a + 2*b)) - (atan((((20*a*b^5 - (((2*a^2*b + a*(-a^3*b)^(1/2) - b*(-a^3*b)^(1/2))/(16*(2*a
^4*b + a^5 + a^3*b^2)))^(1/2)*(128*a^2*b^6 - 64*a*b^7 + 448*a^3*b^5 + 256*a^4*b^4 + tan(c + d*x)*((2*a^2*b + a
*(-a^3*b)^(1/2) - b*(-a^3*b)^(1/2))/(16*(2*a^4*b + a^5 + a^3*b^2)))^(1/2)*(512*a^2*b^7 + 512*a^3*b^6 - 512*a^4
*b^5 - 512*a^5*b^4)) - tan(c + d*x)*(32*a*b^6 + 16*b^7 - 240*a^2*b^5))*((2*a^2*b + a*(-a^3*b)^(1/2) - b*(-a^3*
b)^(1/2))/(16*(2*a^4*b + a^5 + a^3*b^2)))^(1/2) + 4*b^6)*((2*a^2*b + a*(-a^3*b)^(1/2) - b*(-a^3*b)^(1/2))/(16*
(2*a^4*b + a^5 + a^3*b^2)))^(1/2) + 6*b^5*tan(c + d*x))*((2*a^2*b + a*(-a^3*b)^(1/2) - b*(-a^3*b)^(1/2))/(16*(
2*a^4*b + a^5 + a^3*b^2)))^(1/2)*1i - ((20*a*b^5 - (((2*a^2*b + a*(-a^3*b)^(1/2) - b*(-a^3*b)^(1/2))/(16*(2*a^
4*b + a^5 + a^3*b^2)))^(1/2)*(128*a^2*b^6 - 64*a*b^7 + 448*a^3*b^5 + 256*a^4*b^4 - tan(c + d*x)*((2*a^2*b + a*
(-a^3*b)^(1/2) - b*(-a^3*b)^(1/2))/(16*(2*a^4*b + a^5 + a^3*b^2)))^(1/2)*(512*a^2*b^7 + 512*a^3*b^6 - 512*a^4*
b^5 - 512*a^5*b^4)) + tan(c + d*x)*(32*a*b^6 + 16*b^7 - 240*a^2*b^5))*((2*a^2*b + a*(-a^3*b)^(1/2) - b*(-a^3*b
)^(1/2))/(16*(2*a^4*b + a^5 + a^3*b^2)))^(1/2) + 4*b^6)*((2*a^2*b + a*(-a^3*b)^(1/2) - b*(-a^3*b)^(1/2))/(16*(
2*a^4*b + a^5 + a^3*b^2)))^(1/2) - 6*b^5*tan(c + d*x))*((2*a^2*b + a*(-a^3*b)^(1/2) - b*(-a^3*b)^(1/2))/(16*(2
*a^4*b + a^5 + a^3*b^2)))^(1/2)*1i)/(((20*a*b^5 - (((2*a^2*b + a*(-a^3*b)^(1/2) - b*(-a^3*b)^(1/2))/(16*(2*a^4
*b + a^5 + a^3*b^2)))^(1/2)*(128*a^2*b^6 - 64*a*b^7 + 448*a^3*b^5 + 256*a^4*b^4 + tan(c + d*x)*((2*a^2*b + a*(
-a^3*b)^(1/2) - b*(-a^3*b)^(1/2))/(16*(2*a^4*b + a^5 + a^3*b^2)))^(1/2)*(512*a^2*b^7 + 512*a^3*b^6 - 512*a^4*b
^5 - 512*a^5*b^4)) - tan(c + d*x)*(32*a*b^6 + 16*b^7 - 240*a^2*b^5))*((2*a^2*b + a*(-a^3*b)^(1/2) - b*(-a^3*b)
^(1/2))/(16*(2*a^4*b + a^5 + a^3*b^2)))^(1/2) + 4*b^6)*((2*a^2*b + a*(-a^3*b)^(1/2) - b*(-a^3*b)^(1/2))/(16*(2
*a^4*b + a^5 + a^3*b^2)))^(1/2) + 6*b^5*tan(c + d*x))*((2*a^2*b + a*(-a^3*b)^(1/2) - b*(-a^3*b)^(1/2))/(16*(2*
a^4*b + a^5 + a^3*b^2)))^(1/2) + ((20*a*b^5 - (((2*a^2*b + a*(-a^3*b)^(1/2) - b*(-a^3*b)^(1/2))/(16*(2*a^4*b +
a^5 + a^3*b^2)))^(1/2)*(128*a^2*b^6 - 64*a*b^7 + 448*a^3*b^5 + 256*a^4*b^4 - tan(c + d*x)*((2*a^2*b + a*(-a^3
*b)^(1/2) - b*(-a^3*b)^(1/2))/(16*(2*a^4*b + a^5 + a^3*b^2)))^(1/2)*(512*a^2*b^7 + 512*a^3*b^6 - 512*a^4*b^5 -
512*a^5*b^4)) + tan(c + d*x)*(32*a*b^6 + 16*b^7 - 240*a^2*b^5))*((2*a^2*b + a*(-a^3*b)^(1/2) - b*(-a^3*b)^(1/
2))/(16*(2*a^4*b + a^5 + a^3*b^2)))^(1/2) + 4*b^6)*((2*a^2*b + a*(-a^3*b)^(1/2) - b*(-a^3*b)^(1/2))/(16*(2*a^4
*b + a^5 + a^3*b^2)))^(1/2) - 6*b^5*tan(c + d*x))*((2*a^2*b + a*(-a^3*b)^(1/2) - b*(-a^3*b)^(1/2))/(16*(2*a^4*
b + a^5 + a^3*b^2)))^(1/2)))*((2*a^2*b + a*(-a^3*b)^(1/2) - b*(-a^3*b)^(1/2))/(16*(2*a^4*b + a^5 + a^3*b^2)))^
(1/2)*2i)/d - (atan((((20*a*b^5 - (((2*a^2*b - a*(-a^3*b)^(1/2) + b*(-a^3*b)^(1/2))/(16*(2*a^4*b + a^5 + a^3*b
^2)))^(1/2)*(128*a^2*b^6 - 64*a*b^7 + 448*a^3*b^5 + 256*a^4*b^4 + tan(c + d*x)*((2*a^2*b - a*(-a^3*b)^(1/2) +
b*(-a^3*b)^(1/2))/(16*(2*a^4*b + a^5 + a^3*b^2)))^(1/2)*(512*a^2*b^7 + 512*a^3*b^6 - 512*a^4*b^5 - 512*a^5*b^4
)) - tan(c + d*x)*(32*a*b^6 + 16*b^7 - 240*a^2*b^5))*((2*a^2*b - a*(-a^3*b)^(1/2) + b*(-a^3*b)^(1/2))/(16*(2*a
^4*b + a^5 + a^3*b^2)))^(1/2) + 4*b^6)*((2*a^2*b - a*(-a^3*b)^(1/2) + b*(-a^3*b)^(1/2))/(16*(2*a^4*b + a^5 + a
^3*b^2)))^(1/2) + 6*b^5*tan(c + d*x))*((2*a^2*b - a*(-a^3*b)^(1/2) + b*(-a^3*b)^(1/2))/(16*(2*a^4*b + a^5 + a^
3*b^2)))^(1/2)*1i - ((20*a*b^5 - (((2*a^2*b - a*(-a^3*b)^(1/2) + b*(-a^3*b)^(1/2))/(16*(2*a^4*b + a^5 + a^3*b^
2)))^(1/2)*(128*a^2*b^6 - 64*a*b^7 + 448*a^3*b^5 + 256*a^4*b^4 - tan(c + d*x)*((2*a^2*b - a*(-a^3*b)^(1/2) + b
*(-a^3*b)^(1/2))/(16*(2*a^4*b + a^5 + a^3*b^2)))^(1/2)*(512*a^2*b^7 + 512*a^3*b^6 - 512*a^4*b^5 - 512*a^5*b^4)
) + tan(c + d*x)*(32*a*b^6 + 16*b^7 - 240*a^2*b^5))*((2*a^2*b - a*(-a^3*b)^(1/2) + b*(-a^3*b)^(1/2))/(16*(2*a^
4*b + a^5 + a^3*b^2)))^(1/2) + 4*b^6)*((2*a^2*b - a*(-a^3*b)^(1/2) + b*(-a^3*b)^(1/2))/(16*(2*a^4*b + a^5 + a^
3*b^2)))^(1/2) - 6*b^5*tan(c + d*x))*((2*a^2*b - a*(-a^3*b)^(1/2) + b*(-a^3*b)^(1/2))/(16*(2*a^4*b + a^5 + a^3
*b^2)))^(1/2)*1i)/(((20*a*b^5 - (((2*a^2*b - a*(-a^3*b)^(1/2) + b*(-a^3*b)^(1/2))/(16*(2*a^4*b + a^5 + a^3*b^2
)))^(1/2)*(128*a^2*b^6 - 64*a*b^7 + 448*a^3*b^5 + 256*a^4*b^4 + tan(c + d*x)*((2*a^2*b - a*(-a^3*b)^(1/2) + b*
(-a^3*b)^(1/2))/(16*(2*a^4*b + a^5 + a^3*b^2)))^(1/2)*(512*a^2*b^7 + 512*a^3*b^6 - 512*a^4*b^5 - 512*a^5*b^4))
- tan(c + d*x)*(32*a*b^6 + 16*b^7 - 240*a^2*b^5))*((2*a^2*b - a*(-a^3*b)^(1/2) + b*(-a^3*b)^(1/2))/(16*(2*a^4
*b + a^5 + a^3*b^2)))^(1/2) + 4*b^6)*((2*a^2*b - a*(-a^3*b)^(1/2) + b*(-a^3*b)^(1/2))/(16*(2*a^4*b + a^5 + a^3
*b^2)))^(1/2) + 6*b^5*tan(c + d*x))*((2*a^2*b - a*(-a^3*b)^(1/2) + b*(-a^3*b)^(1/2))/(16*(2*a^4*b + a^5 + a^3*
b^2)))^(1/2) + ((20*a*b^5 - (((2*a^2*b - a*(-a^3*b)^(1/2) + b*(-a^3*b)^(1/2))/(16*(2*a^4*b + a^5 + a^3*b^2)))^
(1/2)*(128*a^2*b^6 - 64*a*b^7 + 448*a^3*b^5 + 256*a^4*b^4 - tan(c + d*x)*((2*a^2*b - a*(-a^3*b)^(1/2) + b*(-a^
3*b)^(1/2))/(16*(2*a^4*b + a^5 + a^3*b^2)))^(1/2)*(512*a^2*b^7 + 512*a^3*b^6 - 512*a^4*b^5 - 512*a^5*b^4)) + t
an(c + d*x)*(32*a*b^6 + 16*b^7 - 240*a^2*b^5))*((2*a^2*b - a*(-a^3*b)^(1/2) + b*(-a^3*b)^(1/2))/(16*(2*a^4*b +
a^5 + a^3*b^2)))^(1/2) + 4*b^6)*((2*a^2*b - a*(-a^3*b)^(1/2) + b*(-a^3*b)^(1/2))/(16*(2*a^4*b + a^5 + a^3*b^2
)))^(1/2) - 6*b^5*tan(c + d*x))*((2*a^2*b - a*(-a^3*b)^(1/2) + b*(-a^3*b)^(1/2))/(16*(2*a^4*b + a^5 + a^3*b^2)
))^(1/2)))*((2*a^2*b - a*(-a^3*b)^(1/2) + b*(-a^3*b)^(1/2))/(16*(2*a^4*b + a^5 + a^3*b^2)))^(1/2)*2i)/d

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{a + b \tan ^{4}{\left (c + d x \right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+tan(d*x+c)**4*b),x)

[Out]

Integral(1/(a + b*tan(c + d*x)**4), x)

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